How to Solve Mathematical Induction by tutorciecle123

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									              How to Solve Mathematical Induction
How to Solve Mathematical Induction

One of the most important tasks in mathematics is to discover and characterize regular
patterns or sequences. The main mathematical tool we use to prove statements about
sequences is induction. Induction is a very important tool in computer science for several
reasons, one of which is the fact that a characteristic of most programs is repetition of a
sequence of statements.

To illustrate how induction works, imagine that you are climbing an infinitely high ladder. How
do you know whether you will be able to reach an arbitrarily high rung? Suppose you make
the following two assertions about your climbing abilities:
 1) I can definitely reach the first rung.
 2) Once I get to any rung, I can always climb to the next one up.

If both statements are true, then by statement 1 you can get to the first one, and by statement
2, you can get to the second. By statement 2 again, you can get to the third, and fourth, etc.
Therefore, you can climb as high as you wish. Notice that both of these assertions are
necessary for you to get anywhere on the ladder. If only statement 1 is true, you have no
guarantee of getting beyond the first rung. If only statement 2 is true, you may never be able
to get started.
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Assume that the rungs of the ladder are numbered with the positive integers (1,2,3...). Now
think of a specific property that a number might have. Instead of "reaching an arbitrarily high
rung", we can talk about an arbitrary positive integer having that property. We will use the
shorthand P(n) to denote the positive integer n having property P. How can we use the
ladder-climbing technique to prove that P(n) is true for all positive n? The two assertions we
need to prove are:
 1) P(1) is true
 2) for any positive k, if P(k) is true, then P(k+1) is true

Assertion 1 means we must show the property is true for 1; assertion 2 means that if any
number has property P then so does the next number. If we can prove both of these
statements, then P(n) holds for all positive integers, just as you could climb to an arbitrary
rung of the ladder.

The foundation for arguments of this type is the Principle of Mathematical Induction, which can
be used as a proof technique on statements that have a particular form. We can state it this
way:
A proof by mathematical induction that a proposition P(n) is true for every positive integer
n consists of two steps:

BASE CASE: Show that the proposition P(1) is true.
INDUCTIVE STEP: Assume that P(k) is true for an arbitrarily chosen positive integer k,
and show that under that assumption, P(k+1) must be true.
From these two steps we conclude (by the principle of mathematical induction) that for all
positive integers n, P(n) is true.

Note that we do not prove that P(k) is true (except for k = 1). Instead, we show that if P(k) is
true, then P(k+1) must also be true. That's all that is necessary according to the Principle of
Mathematical Induction. The assumption that P(k) is true is called the induction hypothesis.
Be sure you understand that P(n) and P(k) are not numbers; they are propositions that are
true or false.
                                             Learn More :- Proof Of Mathematical Induction



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As an another illustration, consider an infinite number of dominoes positioned one behind the
other in such a way that if any given one falls, then the one behind it falls too. In order to
establish that the entire chain will fall under a certain set of circumstances, two things are
necessary. First, someone has to push over the first domino; this corresponds to the base
case of induction. Second, we must also know that whenever any domino falls, it is close
enough to the next domino in the chain that it will knock it over. This requirement can be
expressed by saying that whenever domino N falls, so does N+1. This corresponds to using
the induction hypothesis to establish the result for the next value of N. If you ever get
confused about induction, the "domino principle" is a good thing to remember.




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